## Abstract

Consider the holomorphic Hamiltonian action of a compact Lie group K on a compact Kähler manifold M with a moment map Φ:M→k^{∗}. Assume that 0 is a regular value of the moment map. Weitsman raised the question of what we can say about the cohomology of the Kähler quotient M_{0}≔Φ^{−1}(0)∕K if all the ordinary cohomology of M is of type (p,p). In this paper, using the Cartan–Chern–Weil theory we show that in the above context there is a natural surjective Kirwan map from an equivariant version of the Dolbeault cohomology of M onto the Dolbeault cohomology of the Kähler quotient M_{0}. As an immediate consequence, this result provides an answer to the question posed by Weitsman.

Original language | English |
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Pages (from-to) | 43-53 |

Number of pages | 11 |

Journal | Journal of Geometry and Physics |

Volume | 144 |

DOIs | |

State | Published - Oct 2019 |

## Keywords

- Cartan–Chern-Weil theory
- Kähler quotient
- Moment maps